155] 
513 
155. 
A FOURTH MEMOIR UPON QUANTICS. 
[From the Philosophical Transactions of the Royal Society of London, vol. cxlviii. for 
the year 1858, pp. 415—427. Received February 11,—Read March 18, 1858.] 
The object of the present memoir is the further development of the theory of binary 
quantics; it should therefore have preceded so much of my third memoir, t. 147 (1857), 
p. 627, [144], as relates to ternary quadrics and cubics. The paragraphs are numbered 
continuously with those of the former memoirs. The 7 first three paragraphs, Nos. 62 to 64, 
relate to quantics of the general form (*\x, y, ...) m , and they are intended to complete 
the series of definitions and explanations given in Nos. 54 to 61 of my third memoir; 
Nos. 68 to 71, although introduced in reference to binary quantics, relate or may be 
considered as relating to quantics of the like general form. But with these exceptions 
the memoir relates to binary quantics of any order whatever: viz. Nos. 65 to 80 relate 
to the covariants and invariants of the degrees 2, 3 and 4; Nos. 81 and 82 (which are 
introduced somewhat parenthetically) contain the explanation of a process for the 
calculation of the invariant called the Discriminant; Nos. 83 to 85 contain the definitions 
of the Catalecticant, the Lambdaic and the Canonisant, which are functions occurring in 
Professor Sylvester’s theory of the reduction of a binary quantic to its canonical form ; 
and Nos. 86 to 91 contain the definitions of certain covariants or other derivatives 
connected with Bezout’s abbreviated method of elimination, due for the most part to 
Professor Sylvester, and which are called Bezoutiants, Cobezoutiants, &c. I have not in 
the present memoir in any wise considered the theories to which the catalecticant, &c. 
and the last-mentioned other covariants and derivatives relate; the design is to point 
out and precisely define the different covariants or other derivatives which have hitherto 
presented themselves in theories relating to binary quantics, and so to complete, as far 
as may be, the explanation of the terminology of this part of the subject. 
62. If we consider a quantic 
C. II. 
(a, b, ...$#, y, ...) m 
65